Laplace Transformations of Submanifolds

TL;DR

This paper studies the Laplace transformation of submanifolds in Euclidean space, focusing on the properties of the Laplace map and its image for isometric immersions.

Contribution

It provides a fundamental analysis of the Laplace transformations of Euclidean submanifolds, introducing the concept and exploring its properties.

Findings

Defined the Laplace map for isometric immersions.

Analyzed the properties of the Laplace image.

Established foundational results on Laplace transformations.

Abstract

Let $x : M \to E^m$ be an isometric immersion of a Riemannian manifold $M$ into a Euclidean $m$-space. Denote by $\Delta$ the Laplace operator of $M$. Then $\Delta$ gives rise to a differentiable map $L :M \to E^m$, called the Laplace map, defined by $L(p)=(\Delta x)(p)$, $p\in M$. We call $L(M)$ the Laplace image, and the transformation $L :M \to L(M)$ from $M$ onto its Laplace image $L(M)$ the {\it Laplace transformation}. In this monograph, we provide a fundamental study of the Laplace transformations of Euclidean submanifolds.